# American Institute of Mathematical Sciences

March  2012, 32(3): 1011-1046. doi: 10.3934/dcds.2012.32.1011

## Multi-dimensional traveling fronts in bistable reaction-diffusion equations

 1 Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, O-okayama 2-12-1-W8-38, Tokyo 152-8552, Japan

Received  October 2010 Revised  September 2011 Published  October 2011

This paper studies traveling front solutions of convex polyhedral shapes in bistable reaction-diffusion equations including the Allen-Cahn equations or the Nagumo equations. By taking the limits of such solutions as the lateral faces go to infinity, we construct a three-dimensional traveling front solution for any given $g\in C^{\infty}(S^{1})$ with $\min_{0\leq \theta\leq 2\pi}g(\theta)=0$.
Citation: Masaharu Taniguchi. Multi-dimensional traveling fronts in bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 1011-1046. doi: 10.3934/dcds.2012.32.1011
##### References:
 [1] S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening,, Acta. Metall., 27 (1979), 1084. doi: 10.1016/0001-6160(79)90196-2. Google Scholar [2] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics,, Partial Differential Equations and Related Topics, 446 (1975), 5. Google Scholar [3] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics,, Adv. in Math., 30 (1978), 33. doi: 10.1016/0001-8708(78)90130-5. Google Scholar [4] H. Berestycki, P. L. Lions and L. A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in $R^N$,, Indiana Univ. Math. J. \textbf{30} (1981), 30 (1981), 141. doi: 10.1512/iumj.1981.30.30012. Google Scholar [5] J. Buckmaster, Polyhedral flames--an exercise in bimodal bifurcation analysis,, SIAM J. Appl. Math., 44 (1984), 40. doi: 10.1137/0144005. Google Scholar [6] X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations,, Adv. Differential Equations, 2 (1997), 125. Google Scholar [7] X. Chen, J-S. Guo, F. Hamel, H. Ninomiya and J-M. Roquejoffre, Traveling waves with paraboloid like interfaces for balanced bistable dynamics,, Ann. I. H. Poincaré, AN 24 (2007), 369. Google Scholar [8] M. del Pino, M. Kowalczyk and J. Wei, On de Giorgi conjecture in dimension $N\geq9$,, Annals of Math. (to appear)., (). Google Scholar [9] P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions,, Arch. Rat. Mech. Anal., 65 (1977), 335. doi: 10.1007/BF00250432. Google Scholar [10] R. A. Fisher, The advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 355. doi: 10.1111/j.1469-1809.1937.tb02153.x. Google Scholar [11] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'', Springer-Verlag, (1983). Google Scholar [12] F. Hamel, R. Monneau and J.-M. Roquejoffre, Stability of travelling waves in a model for conical flames in two space dimensions,, Ann. Scient. Ec. Norm. Sup. 4ème série, t.37 (2004), 469. Google Scholar [13] F. Hamel, R. Monneau and J.-M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts,, Discrete Contin. Dyn. Syst., 13 (2005), 1069. doi: 10.3934/dcds.2005.13.1069. Google Scholar [14] F. Hamel, R. Monneau and J.-M. Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets,, Discrete Contin. Dyn. Syst., 14 (2006), 75. Google Scholar [15] F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $\mathbbR^N$,, Arch. Rat. Mech. Anal., 157 (2001), 91. doi: 10.1007/PL00004238. Google Scholar [16] F. Hamel and J.-M. Roquejoffre, Heteroclinic connections for multidimensional bistable reaction-diffusion equations,, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 101. doi: 10.3934/dcdss.2011.4.101. Google Scholar [17] M. Haragus and A. Scheel, Corner defects in almost planar interface propagation,, Ann. I. H. Poincaré, AN 23 (2006), 283. Google Scholar [18] Y. I. Kanel', Certain problems on equations in the theory of burning,, Soviet. Math. Dokl., 2 (1961), 48. Google Scholar [19] Y. I. Kanel', Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory,, Mat. Sb. (N.S.), 59 (1962), 245. Google Scholar [20] T. Kapitula, Multidimensional stability of planar travelling waves,, Trans. Amer. Math. Soc., 349 (1997), 257. doi: 10.1090/S0002-9947-97-01668-1. Google Scholar [21] K. Kawasaki and T. Ohta, Kink dynamics in one-dimensional nonlinear systems,, Phys. A, 116 (1982), 573. doi: 10.1016/0378-4371(82)90178-9. Google Scholar [22] Y. Kurokawa and M. Taniguchi, Multi-dimensional pyramidal traveling fronts in the Allen-Cahn equations,, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 1031. doi: 10.1017/S0308210510001253. Google Scholar [23] C. D. Levermore and J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation II,, Comm. Par. Diff. Eq., 17 (1992), 1901. Google Scholar [24] H. Matano, M. Nara and M. Taniguchi, Stability of planar waves in the Allen-Cahn equation,, Comm. Par. Diff. Eq., 34 (2009), 976. Google Scholar [25] J. Nagumo, S. Yoshizawa and S. Arimoto, Bistable transmission lines,, IEEE Trans. Circuit Theory, CT-12 (1965), 400. Google Scholar [26] H. Ninomiya and M. Taniguchi, Traveling curved fronts of a mean curvature flow with constant driving force,, Free boundary problems: Theory and applications I, 13 (2000), 206. Google Scholar [27] H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations,, J. Differential Equations, 213 (2005), 204. doi: 10.1016/j.jde.2004.06.011. Google Scholar [28] H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations,, Discrete Contin. Dyn. Syst., 15 (2006), 819. doi: 10.3934/dcds.2006.15.819. Google Scholar [29] L. A. Peletier and J. Serrin, Uniqueness of positive solutions of semilinear equations in $\mathbfR^n$,, Arch. Rational Mech. Anal., 81 (1983), 181. doi: 10.1007/BF00250651. Google Scholar [30] V. Pérez-Muñuzuri, M. Gómez-Gesteira, A. P. Muñuzuri, V. A. Davydov and V. Pérez-Villar, V-shaped stable nonspiral patterns,, Physical Review E, 51 (1995), 845. Google Scholar [31] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'', Springer-Verlag, (1984). Google Scholar [32] J-M. Roquejoffre and V. Roussier-Michon, Nontrivial large-time behaviour in bistable reaction-diffusion equations,, Annali di Matematica, 188 (2009), 207. Google Scholar [33] D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems,, Indiana Univ. Math. J., 21 (1972), 979. doi: 10.1512/iumj.1972.21.21079. Google Scholar [34] N. Shigesada, K. Kawasaki and E. Teramoto, Traveling periodic waves in heterogeneous environments,, Theoret. Population Biol., 30 (1986), 143. doi: 10.1016/0040-5809(86)90029-8. Google Scholar [35] J. G. Skellam, Random dispersal in theoretical populations,, Biometrika, 38 (1951), 196. Google Scholar [36] F. A. Smith and S. F. Pickering, Bunsen flames of unusual structure,, Proceedings of the Symposium on Combustion, Vol. 1-2 (1948), 1. doi: 10.1016/S1062-2888(65)80006-5. Google Scholar [37] M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations,, SIAM J. Math. Anal., 39 (2007), 319. doi: 10.1137/060661788. Google Scholar [38] M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations,, J. Differential Equations, 246 (2009), 2103. doi: 10.1016/j.jde.2008.06.037. Google Scholar [39] M. Taniguchi, Traveling fronts in perturbed multistable reaction-diffusion equations,, Discrete Contin. Dyn. Syst. - Supplement 2011 (The proceedings for the 8th AIMS International Conference on Dynamical Systems, (2011). Google Scholar [40] J. J. Tyson and P. C. Fife, Target patterns in a realistic model of the Belousov-Zhabotinskii reaction,, J. Chem. Phys., 73 (1980), 2224. doi: 10.1063/1.440418. Google Scholar [41] J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation I,, Comm. Par. Diff. Eq., 17 (1992), 1889. Google Scholar [42] H. Yagisita, Nearly spherically symmetric expanding fronts in a bistable reaction-diffusion equation,, J. Dynam. Differential Equations, 13 (2001), 323. doi: 10.1023/A:1016632124792. Google Scholar

show all references

##### References:
 [1] S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening,, Acta. Metall., 27 (1979), 1084. doi: 10.1016/0001-6160(79)90196-2. Google Scholar [2] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics,, Partial Differential Equations and Related Topics, 446 (1975), 5. Google Scholar [3] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics,, Adv. in Math., 30 (1978), 33. doi: 10.1016/0001-8708(78)90130-5. Google Scholar [4] H. Berestycki, P. L. Lions and L. A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in $R^N$,, Indiana Univ. Math. J. \textbf{30} (1981), 30 (1981), 141. doi: 10.1512/iumj.1981.30.30012. Google Scholar [5] J. Buckmaster, Polyhedral flames--an exercise in bimodal bifurcation analysis,, SIAM J. Appl. Math., 44 (1984), 40. doi: 10.1137/0144005. Google Scholar [6] X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations,, Adv. Differential Equations, 2 (1997), 125. Google Scholar [7] X. Chen, J-S. Guo, F. Hamel, H. Ninomiya and J-M. Roquejoffre, Traveling waves with paraboloid like interfaces for balanced bistable dynamics,, Ann. I. H. Poincaré, AN 24 (2007), 369. Google Scholar [8] M. del Pino, M. Kowalczyk and J. Wei, On de Giorgi conjecture in dimension $N\geq9$,, Annals of Math. (to appear)., (). Google Scholar [9] P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions,, Arch. Rat. Mech. Anal., 65 (1977), 335. doi: 10.1007/BF00250432. Google Scholar [10] R. A. Fisher, The advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 355. doi: 10.1111/j.1469-1809.1937.tb02153.x. Google Scholar [11] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'', Springer-Verlag, (1983). Google Scholar [12] F. Hamel, R. Monneau and J.-M. Roquejoffre, Stability of travelling waves in a model for conical flames in two space dimensions,, Ann. Scient. Ec. Norm. Sup. 4ème série, t.37 (2004), 469. Google Scholar [13] F. Hamel, R. Monneau and J.-M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts,, Discrete Contin. Dyn. Syst., 13 (2005), 1069. doi: 10.3934/dcds.2005.13.1069. Google Scholar [14] F. Hamel, R. Monneau and J.-M. Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets,, Discrete Contin. Dyn. Syst., 14 (2006), 75. Google Scholar [15] F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $\mathbbR^N$,, Arch. Rat. Mech. Anal., 157 (2001), 91. doi: 10.1007/PL00004238. Google Scholar [16] F. Hamel and J.-M. Roquejoffre, Heteroclinic connections for multidimensional bistable reaction-diffusion equations,, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 101. doi: 10.3934/dcdss.2011.4.101. Google Scholar [17] M. Haragus and A. Scheel, Corner defects in almost planar interface propagation,, Ann. I. H. Poincaré, AN 23 (2006), 283. Google Scholar [18] Y. I. Kanel', Certain problems on equations in the theory of burning,, Soviet. Math. Dokl., 2 (1961), 48. Google Scholar [19] Y. I. Kanel', Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory,, Mat. Sb. (N.S.), 59 (1962), 245. Google Scholar [20] T. Kapitula, Multidimensional stability of planar travelling waves,, Trans. Amer. Math. Soc., 349 (1997), 257. doi: 10.1090/S0002-9947-97-01668-1. Google Scholar [21] K. Kawasaki and T. Ohta, Kink dynamics in one-dimensional nonlinear systems,, Phys. A, 116 (1982), 573. doi: 10.1016/0378-4371(82)90178-9. Google Scholar [22] Y. Kurokawa and M. Taniguchi, Multi-dimensional pyramidal traveling fronts in the Allen-Cahn equations,, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 1031. doi: 10.1017/S0308210510001253. Google Scholar [23] C. D. Levermore and J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation II,, Comm. Par. Diff. Eq., 17 (1992), 1901. Google Scholar [24] H. Matano, M. Nara and M. Taniguchi, Stability of planar waves in the Allen-Cahn equation,, Comm. Par. Diff. Eq., 34 (2009), 976. Google Scholar [25] J. Nagumo, S. Yoshizawa and S. Arimoto, Bistable transmission lines,, IEEE Trans. Circuit Theory, CT-12 (1965), 400. Google Scholar [26] H. Ninomiya and M. Taniguchi, Traveling curved fronts of a mean curvature flow with constant driving force,, Free boundary problems: Theory and applications I, 13 (2000), 206. Google Scholar [27] H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations,, J. Differential Equations, 213 (2005), 204. doi: 10.1016/j.jde.2004.06.011. Google Scholar [28] H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations,, Discrete Contin. Dyn. Syst., 15 (2006), 819. doi: 10.3934/dcds.2006.15.819. Google Scholar [29] L. A. Peletier and J. Serrin, Uniqueness of positive solutions of semilinear equations in $\mathbfR^n$,, Arch. Rational Mech. Anal., 81 (1983), 181. doi: 10.1007/BF00250651. Google Scholar [30] V. Pérez-Muñuzuri, M. Gómez-Gesteira, A. P. Muñuzuri, V. A. Davydov and V. Pérez-Villar, V-shaped stable nonspiral patterns,, Physical Review E, 51 (1995), 845. Google Scholar [31] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'', Springer-Verlag, (1984). Google Scholar [32] J-M. Roquejoffre and V. Roussier-Michon, Nontrivial large-time behaviour in bistable reaction-diffusion equations,, Annali di Matematica, 188 (2009), 207. Google Scholar [33] D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems,, Indiana Univ. Math. J., 21 (1972), 979. doi: 10.1512/iumj.1972.21.21079. Google Scholar [34] N. Shigesada, K. Kawasaki and E. Teramoto, Traveling periodic waves in heterogeneous environments,, Theoret. Population Biol., 30 (1986), 143. doi: 10.1016/0040-5809(86)90029-8. Google Scholar [35] J. G. Skellam, Random dispersal in theoretical populations,, Biometrika, 38 (1951), 196. Google Scholar [36] F. A. Smith and S. F. Pickering, Bunsen flames of unusual structure,, Proceedings of the Symposium on Combustion, Vol. 1-2 (1948), 1. doi: 10.1016/S1062-2888(65)80006-5. Google Scholar [37] M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations,, SIAM J. Math. Anal., 39 (2007), 319. doi: 10.1137/060661788. Google Scholar [38] M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations,, J. Differential Equations, 246 (2009), 2103. doi: 10.1016/j.jde.2008.06.037. Google Scholar [39] M. Taniguchi, Traveling fronts in perturbed multistable reaction-diffusion equations,, Discrete Contin. Dyn. Syst. - Supplement 2011 (The proceedings for the 8th AIMS International Conference on Dynamical Systems, (2011). Google Scholar [40] J. J. Tyson and P. C. Fife, Target patterns in a realistic model of the Belousov-Zhabotinskii reaction,, J. Chem. Phys., 73 (1980), 2224. doi: 10.1063/1.440418. Google Scholar [41] J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation I,, Comm. Par. Diff. Eq., 17 (1992), 1889. Google Scholar [42] H. Yagisita, Nearly spherically symmetric expanding fronts in a bistable reaction-diffusion equation,, J. Dynam. Differential Equations, 13 (2001), 323. doi: 10.1023/A:1016632124792. Google Scholar
 [1] Fang Li, Kimie Nakashima. Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1391-1420. doi: 10.3934/dcds.2012.32.1391 [2] Hirokazu Ninomiya. Entire solutions and traveling wave solutions of the Allen-Cahn-Nagumo equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2001-2019. doi: 10.3934/dcds.2019084 [3] Hongmei Cheng, Rong Yuan. Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1015-1029. doi: 10.3934/dcdsb.2015.20.1015 [4] Jiao Chen, Weike Wang. The point-wise estimates for the solution of damped wave equation with nonlinear convection in multi-dimensional space. Communications on Pure & Applied Analysis, 2014, 13 (1) : 307-330. doi: 10.3934/cpaa.2014.13.307 [5] Grégory Faye. Multidimensional stability of planar traveling waves for the scalar nonlocal Allen-Cahn equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2473-2496. doi: 10.3934/dcds.2016.36.2473 [6] Luyi Ma, Hong-Tao Niu, Zhi-Cheng Wang. Global asymptotic stability of traveling waves to the Allen-Cahn equation with a fractional Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2457-2472. doi: 10.3934/cpaa.2019111 [7] Gianni Gilardi. On an Allen-Cahn type integrodifferential equation. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 703-709. doi: 10.3934/dcdss.2013.6.703 [8] Isabeau Birindelli, Enrico Valdinoci. On the Allen-Cahn equation in the Grushin plane: A monotone entire solution that is not one-dimensional. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 823-838. doi: 10.3934/dcds.2011.29.823 [9] Jun Yang, Xiaolin Yang. Clustered interior phase transition layers for an inhomogeneous Allen-Cahn equation in higher dimensional domains. Communications on Pure & Applied Analysis, 2013, 12 (1) : 303-340. doi: 10.3934/cpaa.2013.12.303 [10] Jean-Paul Chehab, Alejandro A. Franco, Youcef Mammeri. Boundary control of the number of interfaces for the one-dimensional Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 87-100. doi: 10.3934/dcdss.2017005 [11] Hirokazu Ninomiya, Masaharu Taniguchi. Global stability of traveling curved fronts in the Allen-Cahn equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 819-832. doi: 10.3934/dcds.2006.15.819 [12] Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127 [13] Xinlong Feng, Huailing Song, Tao Tang, Jiang Yang. Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inverse Problems & Imaging, 2013, 7 (3) : 679-695. doi: 10.3934/ipi.2013.7.679 [14] Christos Sourdis. On the growth of the energy of entire solutions to the vector Allen-Cahn equation. Communications on Pure & Applied Analysis, 2015, 14 (2) : 577-584. doi: 10.3934/cpaa.2015.14.577 [15] Paul H. Rabinowitz, Ed Stredulinsky. On a class of infinite transition solutions for an Allen-Cahn model equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 319-332. doi: 10.3934/dcds.2008.21.319 [16] Ciprian G. Gal, Maurizio Grasselli. The non-isothermal Allen-Cahn equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 1009-1040. doi: 10.3934/dcds.2008.22.1009 [17] Eleonora Cinti. Saddle-shaped solutions for the fractional Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 441-463. doi: 10.3934/dcdss.2018024 [18] Zhuoran Du, Baishun Lai. Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1407-1429. doi: 10.3934/dcds.2013.33.1407 [19] Charles-Edouard Bréhier, Ludovic Goudenège. Analysis of some splitting schemes for the stochastic Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4169-4190. doi: 10.3934/dcdsb.2019077 [20] Yoshihiro Ueda, Tohru Nakamura, Shuichi Kawashima. Stability of planar stationary waves for damped wave equations with nonlinear convection in multi-dimensional half space. Kinetic & Related Models, 2008, 1 (1) : 49-64. doi: 10.3934/krm.2008.1.49

2018 Impact Factor: 1.143